Direct characterization of finite-dimensional $1$-injective Banach spaces

Question

It follows from Kelley's Theorem that the only finite-dimensional $1$-injective Banach spaces are $\ell^\infty_n$, $n\in\mathbb N$. Is there a simple direct proof of this fact, without having to talk about order-complete spaces of continuous functions? Is there such proof for $c_0$ or $\ell^p_n$ with $p<\infty$, say?

A possible strategy could be to somehow embed $\ell^p_n$ (or any other finite-dimensional Banach space, not isometrically isomorphic to $\ell^\infty_n$) isometrically as a direct summand some well-chosen Banach space $\mathcal X$ and show that any bounded projection onto $\ell^p_n$ has norm greater than $1$. My naive attempts at this did not bear fruit for it amounts to norming $\ell^p_n\oplus \mathcal Y$ in some way such that it restricts to the $p$-norm in the first component (there are many ways to do this) but then one would have to calculate the norm of an arbitrary projection onto the first component, that is maps of the form $P(x,y)=(x+Ty,0)$ for $T:\mathcal Y\to\ell^p_n$ bounded, and there I got stuck. I don't know if this idea can be saved, or some other strategy works.

Answer 1

There are no really simple proofs of the characterization of finite dimensional $1$-injective spaces, but there are various proofs of even generalizations that do not directly rely on the Kelley-Goodner-Nachbin proofs. One is in section 4 of my 1980 paper with Figiel, LARGE SUBSPACES OF $\ell_\infty^n$ AND ESTIMATES OF THE GORDON-LEWIS CONSTANT, Israel J. Math. 37, and another in the paper by J.-P. Deschaseaux that is referenced in our paper. Another approach is to prove the equivalent dual version; namely, that a contractively complemented subspace of $L_1$ of dimesion $n$ must be isometrically isomorphic to $\ell_1^n$. Zippin even proved a version of this for fintite dimensional $1+\epsilon$-complemented subspaces of $L_1$.